Geiringer velocity relations ($du-vd\varphi =0\text{on }\alpha ,dv+ud\varphi =0\text{on }\beta$)

For an associated flow rule in the reduced plane-strain yield coordinates, the rate-of-strain tensor has zero extension along both slip line directions. Write velocity as $u{e}_{\alpha }+v{e}_{\beta }$, with the local orthonormal axes turning through angle $\varphi$. Differentiating along an $\alpha$ curve gives longitudinal extension $du-vd\varphi$; along a $\beta$ curve it gives $dv+ud\varphi$. Setting these extensions to zero proves the relations for the corresponding anisotropic yield surface parameterization.